few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Solve real-world and mathematical problems involving area, surface area, and volume.

Analyze, compare, create, and compose shapes.

Experiment with transformations in the plane.

Explain volume formulas and use them to solve problems.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Use coordinates to prove simple geometric theorems algebraically.

Apply trigonometry to general triangles.

Understand and apply theorems about circles.

Apply geometric concepts in modeling situations.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Prove theorems involving similarity.

Define trigonometric ratios and solve problems involving right triangles.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Classify two-dimensional figures into categories based on their properties.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Understand congruence in terms of rigid motions.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Find arc lengths and areas of sectors of circles.

Understand similarity in terms of similarity transformations.

Understand and apply the Pythagorean Theorem.

Translate between the geometric description and the equation for a conic section.

Reason with shapes and their attributes.

Make geometric constructions.

Prove geometric theorems.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Visualize relationships between two-dimensional and three-dimensional objects.
problem_activity: Activity
Students continue to practice drawing triangles from given conditions and categorizing their results. This activity focuses on the inclusion of a single angle and two sides. Again, they do not need to memorize which conditions result in unique triangles, but should begin to notice how some conditions (such as the equal side lengths) result in certain requirements for the completed triangle.
There is an optional blackline master that can help students organize their work at trying different configurations of the first set of measurements. If you provide students with a copy of the blackline master, ask them to determine whether any of the configurations result in the same triangle, as well as whether any one configuration results in two possible triangles.
Digital
Launch
Keep students in same groups. Remind students of the activity in a previous lesson where they used the strips and fasteners to draw triangles on their paper. Ask what other tool also helps you find all the points that are a certain distance from a center point (a compass). Distribute optional blackline masters if desired. Provide access to geometry toolkits and compasses.
Give students 7–8 minutes of partner work time, followed by a whole-class discussion.
If students have access to digital activities there is an applet that allows for triangle construction.
Reading, Speaking: MLR5 Co-craft Questions.
Display just the statement: “One angle measures 40 degrees, one side measures 4 cm, and one side measures 5 cm.” Invite students to write down possible mathematical questions that could be asked with this information. Ask students to compare the questions generated with a partner before selecting 1–2 groups to share their questions with the class. Listen for the ways the given conditions are used or referenced in students’ questions. This routine will help students to understand the context of this problem before they are asked to create a drawing.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
Use the applet to draw triangles.
Draw as many different triangles as you can with each of these sets of measurements:
One angle measures
\(40^\circ\)
, one side measures 4 cm and one side measures 5 cm.
Two sides measure 6 cm and one angle measures
\(100^\circ\)
.
Did either of these sets of measurements determine one unique triangle? How do you know?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Keep students in same groups. Remind students of the activity in a previous lesson where they used the strips and fasteners to draw triangles on their paper. Ask what other tool also helps you find all the points that are a certain distance from a center point (a compass). Distribute optional blackline masters if desired. Provide access to geometry toolkits and compasses.
Give students 7–8 minutes of partner work time, followed by a whole-class discussion.
If students have access to digital activities there is an applet that allows for triangle construction.
Reading, Speaking: MLR5 Co-craft Questions.
Display just the statement: “One angle measures 40 degrees, one side measures 4 cm, and one side measures 5 cm.” Invite students to write down possible mathematical questions that could be asked with this information. Ask students to compare the questions generated with a partner before selecting 1–2 groups to share their questions with the class. Listen for the ways the given conditions are used or referenced in students’ questions. This routine will help students to understand the context of this problem before they are asked to create a drawing.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
Draw as many different triangles as you can with each of these sets of measurements:
One angle measures
\(40^\circ\)
, one side measures 4 cm, and one side measures 5 cm.
Two sides measure 6 cm, and one angle measures
\(100^\circ\)
.
Did either of these sets of measurements determine one unique triangle? How do you know?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may draw two different orientations of the same triangle for the first set of conditions, with the
\(40^\circ\)
angle in between the 4 cm and 5 cm sides. Prompt them to use tracing paper to check whether their two triangles are really different (not identical copies).
If students struggle to create more than one triangle from the first set of conditions, prompt them to write down the order they already used for their measurements and then brainstorm other possible orders they could use.
Activity Synthesis
Ask one or more students to share how many different triangles they were able to draw with each set of conditions. Select students to share their solutions.
If not brought up in student explanations, point out that for the first problem, one possible order for the measurements (
\(40^\circ\)
, 5 cm, 4 cm) can result in two different triangles (the bottom two in the solution image). One way to show this is to draw a 5 cm segment and then use a compass to draw a circle with a 4 cm radius centered on the segment’s left endpoint. Next, draw a ray at a
\(40^\circ\)
angle centered on the segment’s right endpoint. Notice that this ray intersects the circle twice. Each one of these points could be the third vertex of the triangle. While it is helpful for students to notice this interesting aspect of their drawing, it is
not
important for students to learn rules about the number of possible triangles given different sets of conditions.
If the optional blackline master was used, ask students:
“Which configurations made identical triangles?” (the top left and bottom left)
“Which configurations made more than one triangle?” (the bottom right)
If not mentioned by students, explain to students that the top left and bottom left configurations result in the same triangle, because in both cases the
\(40^\circ\)
angle is in between the 4 cm and 5 cm sides and that the bottom right configuration results in two different triangles, because the arc intersects the ray in two different places.
MLR 1 (Stronger and Clearer Each Time): Before discussing the second set of conditions as a whole class, have student pairs share their reasoning for why there were no more triangles that could be drawn with the given measures, with two different partners in a row. Have students practice using mathematical language to be as clear as possible when sharing with the class, when and if they are called upon.
Engagement: Develop Effort and Persistence.
Break the class into small group discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Attention; Social-emotional skills


correct_option_index: 25